Question

The Setup: A piece of wire of length 10 cm is cut into two (not necessarily equal length) pieces. One piece of length x cm is made into a circle and the rest is made into a square.

1. Express the sum of the areas of the square and circle as a function of x

2. For what x-value is maximum area achieved?

3. For maximum area, how should the wire be bent?

4. Find the critical values of your function f[x].

5. Now give the radius and side length of the circle and square for which area is
minimized.

6. A curious property of the circle and square that maximize or minimize volume follows. For the x-values you computed that minimized and maximized the combined areas of the circle and the square, have Mathematica verify the following for you:
Length of wire used for the square/Length of wire used for the circle = area of square/area of circle
Use exact values and the simplify command to verify that both ratios are the same.

7. You might wonder if there are are other values of x for which,
Length of wire used for the square/Length of wire used for the circle = area of square/area of circle
Use the Solve[ ] command in Mathematica by solving this proportion for x, where x is the circumference of the circle as before. What do you notice about the solutions to this equation and the values of x that minimized and maximized the combined areas?

Answer 1